Text uv layer texture
The Material

A material is set up for the object, coloring an emission shader with an image in the object's UV space, through a Mapping node:



Unwrapping

If you're following a method something like this one, then you're just unwrapping flat surfaces, before they are modified:




There is a seam down every longitudinal edge, so it's just U unwrap, followed by rotating/scaling in the UV Editor.
Although the UV space has been set up to allow different text on every face, in this case the UV islands have been snapped over one another, so they all refer to the same region of the image.


Modifiers

After Twist and Bend modifiers are assigned as shown in the linked answer ..



Animation

Now the Location: X field of the Mapping node in the material can be keyframed between 0 and 1 to loop the translation of the UV texture space (trim the last  frame, where X = 1 = 0, to get a seamless loop):

Here, the loop is shown without the animated twist: only the text is being animated across its surface.



Here, with the twist as well..